论文标题
表面上的组合随机曲线
Combinatorially random curves on surfaces
论文作者
论文摘要
我们研究了在欧拉的负面特征的可定向表面$ s $上随机封闭曲线的拓扑特性。让$γ_{n} $表示$ n^{th} $步骤的共轭类别的简单随机步行的步骤，该步骤是由由一项措施驱动的cayley图表上的，该措施的支撑在有限的生成集上，然后概率收敛到$ 1 $ as $ n $ as $ n $，（$ n $ （2）$γ_{n} $的自我交流数为$ n^{2} $，所有双曲线指标的最小长度为$γ_{n} $，按$ n $订单为$ n $，而公制最小的长度最小为$γ_{n} $均匀的厚度； （3）当$ s $被刺穿并且分布是均匀的，并在最小尺寸的生成集中支撑时，$γ_{n} $的最低封面程度承认了一个简单的高度（我们称之为$ \ textit {简单提升度} $ $γ_{n} $）至少在$ n/$ n/$ n）上（$ n/\ n），并且在$ n/\ n）上（n log n oc n n ocy n oc n n/\ n）and（和n log n oc n log and $ n）and \ n）and \ n）and \ n）and \ n）and \ n）。 我们还表明，这些属性是$ \ textit {generic} $，从某种意义上说，在cayley图中，radius $ n $中的元素比例的比例将其固定为$ 1 $ a $ a $ n $ comlement to $ n $ comply。 对于随机选择的曲线的简单提升度的下限，我们获得了明显改善以前最佳的界限，该界限按$ \ log^{（1/3）} n $处于$ \ log^{（1/3）} n $。作为应用，我们在其定义曲线的自我交流数以及在其相互作用数量的随机曲线的简单提升程度上，在其相互作用数的随机曲线上，在其相互作用数量的随机曲线上，在其相互作用数的随机曲线上，对通用的伪 - Anosov同态的扩张提供了相对较清晰的上限和下限。
We study topological properties of random closed curves on an orientable surface $S$ of negative Euler characteristic. Letting $γ_{n}$ denote the conjugacy class of the $n^{th}$ step of a simple random walk on the Cayley graph driven by a measure whose support is on a finite generating set, then with probability converging to $1$ as $n$ goes to infinity, (1) the point in Teichmüller space at which $γ_{n}$ is length-minimized stays in some compact set; (2) the self-intersection number of $γ_{n}$ is on the order of $n^{2}$, the minimum length of $γ_{n}$ taken over all hyperbolic metrics is on the order of $n$, and the metric minimizing the length of $γ_{n}$ is uniformly thick; and (3) when $S$ is punctured and the distribution is uniform and supported on a generating set of minimum size, the minimum degree of a cover to which $γ_{n}$ admits a simple elevation (which we call the $\textit{simple lifting degree}$ of $γ_{n}$) grows at least like $n/\log(n)$ and at most on the order of $n$. We also show that these properties are $\textit{generic}$, in the sense that the proportion of elements in the ball of radius $n$ in the Cayley graph for which they hold, converges to $1$ as $n$ goes to infinity. The lower bounds on simple lifting degree for randomly chosen curves we obtain significantly improve the previously best known bounds which were on the order of $\log^{(1/3)}n$. As applications, we give relatively sharp upper and lower bounds on the dilatation of a generic point-pushing pseudo-Anosov homeomorphism in terms of the self-intersection number of its defining curve, as well as upper bounds on the simple lifting degree of a random curve in terms of its intersection number which outperform bounds for general curves.
